5.7. BOUNDARY-LAYER FLOW AND TURBULENCE IN HEAT TRANSFER

5.7A. Laminar Flow and Boundary-Layer Theory in Heat Transfer

In Section 3.10C an exact solution was obtained for the velocity profile for isothermal laminar flow past a flat plate. The solution of Blasius can be extended to include the convective heat-transfer problem for the same geometry and laminar flow. In Fig. 5.7-1 the thermal boundary layer is shown. The temperature of the fluid approaching the plate is T and that of the plate is TS at the surface.

Figure 5.7-1. Laminar flow of fluid past a flat plate and thermal boundary layer.


We start by writing the differential energy balance, Eq. (5.6-13):

Equation 5.7-1


If the flow is in the x and y directions, νz = 0. At steady state, ∂T/∂t = 0. Conduction is neglected in the x and z directions, so ∂2T/∂x2 = ∂2T/∂z2 = 0. Conduction occurs in the y direction. The result is

Equation 5.7-2


The simplified momentum-balance equation used in the velocity boundary-layer derivation is very similar:

Equation 3.10-5


The continuity equation used previously is

Equation 3.10-3


Equations (3.10-5) and (3.10-3) were used by Blasius for solving the case for laminar boundary-layer flow. The boundary conditions used were

Equation 5.7-3


The similarity between Eqs. (3.10-5) and (5.7-2) is obvious. Hence, the Blasius solution can be applied if k/ρcp = μ/ρ. This means the Prandtl number cpμ/k = 1. Also, the boundary conditions must be the same. This is done by replacing the temperature T in Eq. (5.7-2) by the dimensionless variable (TTS)/(TTS). The boundary conditions become

Equation 5.7-4


We see that the equations and boundary conditions are identical for the temperature profile and the velocity profile. Hence, for any point x, y in the flow system, the dimensionless velocity variables νx/ν and (TTS)/(TTS) are equal. The velocity-profile solution is the same as the temperature-profile solution.

This means that the transfer of momentum and heat are directly analogous, and the boundary-layer thickness δ for the velocity profile (hydrodynamic boundary layer) and the thermal boundary-layer thickness δT are equal. This is important for gases, where the Prandtl numbers are close to 1.

By combining Eqs. (3.10-7) and (3.10-8), the velocity gradient at the surface is

Equation 5.7-5


where NRe,x = ρ/μ. Also,

Equation 5.7-6


Combining Eqs. (5.7-5) and (5.7-6),

Equation 5.7-7


The convective equation can be related to the Fourier equation by the following, where qy is in J/s or W (btu/h):

Equation 5.7-8


Combining Eqs. (5.7-7) and (5.7-8),

Equation 5.7-9


where NNu,x is the dimensionless Nusselt number and hx is the local heat-transfer coefficient at point x on the plate.

Pohlhausen (K1) was able to show that the relation between the hydrodynamic and thermal boundary layers for fluids with Prandtl number >0.6 gives approximately

Equation 5.7-10


As a result, the equation for the local heat-transfer coefficient is

Equation 5.7-11


Also,

Equation 5.7-12


The equation for the mean heat-transfer coefficient h from x = 0 to x = L for a plate of width b and area bL is

Equation 5.7-13


Integrating,

Equation 5.7-14


Equation 5.7-15


As pointed out previously, this laminar boundary layer on smooth plates holds up to a Reynolds number of about 5 × 105. In using the results above, the fluid properties are usually evaluated at the film temperature Tf = (TS + T)/2.

5.7B. Approximate Integral Analysis of the Thermal Boundary Layer

As discussed in the analysis of the hydrodynamic boundary layer, the Blasius solution is accurate but limited in its scope. Other, more complex systems cannot be solved by this method. The approximate integral analysis used by von Kármán to calculate the hydrodynamic boundary layer was covered in Section 3.10. This approach can be used to analyze the thermal boundary layer.

This method will be outlined briefly. First, a control volume, as previously given in Fig. 3.10-5, is used to derive the final energy integral expression:

Equation 5.7-16


This equation is analogous to Eq. (3.10-48) combined with Eq. (3.10-51) for the momentum analysis, giving

Equation 5.7-17


Equation (5.7-16) can be solved if both a velocity profile and temperature profile are known. The assumed velocity profile used is Eq. (3.10-50):

Equation 3.10-50


The same form of temperature profile is assumed:

Equation 5.7-18


Substituting Eqs. (3.10-50) and (5.7-18) into the integral expression and solving,

Equation 5.7-19


This is only about 8% greater than the exact result in Eq. (5.7-11), which indicates that this approximate integral method can be used with confidence in cases where exact solutions cannot be obtained.

In a similar fashion, the integral momentum analysis method used for the turbulent hydrodynamic boundary layer in Section 3.10 can be used for the thermal boundary layer in turbulent flow. Again, the Blasius -power law is used for the temperature distribution. These give results that are quite similar to the experimental equations given in Section 4.6.

5.7C. Prandtl Mixing Length and Eddy Thermal Diffusivity

1. Eddy momentum diffusivity in turbulent flow

In Section 3.10F the total shear stress for turbulent flow was written as follows when the molecular and turbulent contributions are summed together:

Equation 5.7-20


The molecular momentum diffusivity μ/ρ in m2/s is a function only of the fluid molecular properties. However, the turbulent momentum eddy diffusivity εt depends on the fluid motion. In Eq. (3.10-29) we related εt to the Prandtl mixing length L as follows:

Equation 3.10-29


2. Prandtl mixing length and eddy thermal diffusivity

We can derive the eddy thermal diffusivity αt for turbulent heat transfer in a similar manner, as follows. Eddies or clumps of fluid are transported a distance L in the y direction. At this point L the clump of fluid differs in mean velocity from the adjacent fluid by the velocity , which is the fluctuating velocity component discussed in Section 3.10F. Energy is also transported the distance L with a velocity in the y direction together with the mass being transported. The instantaneous temperature of the fluid is where is the mean value and the deviation from the mean value. This fluctuating is similar to the fluctuating velocity . The mixing length is small enough that the temperature difference can be written as

Equation 5.7-21


The rate of energy transported per unit area is qy/A and is equal to the mass flux in the y direction times the heat capacity times the temperature difference:

Equation 5.7-22


In Section 3.10F we assumed and that

Equation 5.7-23


Substituting Eq. (5.7-23) into (5.7-22),

Equation 5.7-24


According to Eq. (3.10-29) the term is the momentum eddy diffusivity εt. When this term is in the turbulent heat-transfer equation (5.7-24), it is called αt, eddy thermal diffusivity. Then Eq. (5.7-24) becomes

Equation 5.7-25


Combining this with the Fourier equation written in terms of the molecular thermal diffusivity α,

Equation 5.7-26


3. Similarities among momentum, heat, and mass transport

Equation (5.7-26) is similar to Eq. (5.7-20) for total momentum transport. The eddy thermal diffusivity αt and the eddy momentum diffusivity εt have been assumed equal in the derivations. Experimental data show that this equality is only approximate. An eddy mass diffusivity for mass transfer has been defined in a similar manner using the Prandtl mixing length theory and is assumed equal to αt and εt.

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